Chase And Trim (CAT): Active One-Rotation Deflation for the Symmetric QR Eigenvalue Algorithm
Year: 2026
Type: preprint
Abstract: Chase And Trim (CAT) augments the symmetric tridiagonal QR eigenvalue algorithm with active one-rotation deflation. When a first-band coupling is not yet small enough for conventional deflation, CAT tests whether a Jacobi similarity rotation can annihilate it while creating only negligible cross-cut fill. If the resulting fill entries lie within the same local tolerance used for ordinary deflation, they are discarded and the active block is split earlier into independent subproblems. We derive the exact split condition and a cheaper sufficient acceptance inequality, organize non-intersecting candidate cuts for parallel evaluation and execution, and analyze the resulting arithmetic work and dependency span. The active-split perturbation satisfies the same local spectral-norm tolerance bound used for conventional deflation. Across eight tested matrix families and four matrix sizes, CAT reduced the number of QR sweep positions in all 32 matrix-family/size combinations, with reductions of approximately 2%–21%. Weighted arithmetic span, the main metric in the parallel model, was also reduced in all 32 combinations, by approximately 1.5%–28%. Weighted arithmetic work decreased in 23 of 32 combinations. Across the tested cases, the largest combined CAT overhead was about 4.3% of weighted work and 1.0% of weighted span. Candidate and baseline eigenvalue errors relative to LAPACK remained of the same order. The study reports algorithmic work-and-span results rather than wall-clock performance.
Authors: Byazrova, Eteri, Morozov, Yuri
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214012959
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